Mixed toric residues and tropical degenerations

Building on our earlier work on toric residues and reduction, we give a proof for the mixed toric residue conejecture of Batyrev and Materov. We simplify and streamline our technique of tropical degenerations, which allows one to interpolate between two localization principles: one appearing in the intersection theory toric quotients and the other in the calculus of toric residues. This quickly leads to the proof of the conjecture, which gives a closed form for the summation of a generating series whose coefficients are certain variants of the "numbers" of rational curves on toric complete intersection Calabi-Yau manifolds.